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Mathematics > Combinatorics

Title:
On the Number of Acyclic Orientations of Complete $k$-Partite Graphs

Abstract: Building on previous work by Cameron et al. in [3], we give a recurrence for
computing the number of acyclic orientations of complete $k$-partite graphs,
which can be implemented to obtain a dynamic programming algorithm running in
time $n^{O(k)}$, where $n$ is the number of vertices in the graph. We prove our
result by using a relationship between the number of acyclic orientations and
the number of Hamiltonian paths in complete $k$-partite graphs and providing a
recurrence for the latter quantity. We give a simple extension of our algorithm
to the situation when we are an edge removal away from having a complete
$k$-partite graph.